NON-COMMUTATIVE DUPLICATES OF FINITE SETS
CLAUDE CIBILS
Abstract. We classify the twisted tensor products of a finite set algebra with
a two elements set algebra using colored quivers obtained through considerations analogous to Ore extensions. The resulting 2-nilpotent algebras have
particular features with respect to Hochschild (co)homology and cyclic homology.
1. Introduction
A geometric category can be studied through the category of commutative algebras of functions on the spaces with values in a a commutative ring or a field,
where the functions have properties according to the geometrical nature of the
spaces; both categories are usually equivalent through this contravariant functor.
Using this equivalence the algebraic approach is a substitute for the geometric one,
keeping the geometric origine as a background. In this context the tensor product of
algebras corresponds to the product of the spaces. The non-commutative geometry
approach uses non-commutative algebras as substitutes for the spaces, providing a
setting corresponding to virtual non-commutative spaces.
The set algebra of k valued functions over a a finite set E is a commutative
semi-simple basic finite dimensional k-algebra, isomorphic to k × · · · × k where the
number of copies of k equals the cardinality of E. The tensor product of algebras
k E ⊗k k F corresponds to the cartesian product E × F . The non-commutative
counterpart consist on twisted tensor products of k E and k F corresponding to noncommutative virtual cartesian products of the sets E and F . Recall that a twisted
tensor product of two algebras is an algebra structure on the tensor product of the
underlying vector spaces such that the natural inclusions of the original algebras are
algebra maps. This structure has been considered by P. Cartier when lecturing in
Paris, [3]. Several authors have also considered such product in different contexts,
see the work of J. Baez [1], A. Cap, H. Schichl, J. Vanžura [2] and G. Maltsiniotis
[6]. Note that twisted tensor products between an algebra and the polynomial
algebra in one variable corresponds precisely to Ore extensions, see for instance [7]
for the definition and use of Ore extensions.
The problem of classifying structures arising this way is difficult. Quite diverse
non semi-simple one parameter families of algebras can occur. In this paper we use
the equivalence between finite sets and semi-simple basic algebras over a field k in
order to classify the non-commutative duplicates of a set. More precisely the usual
duplicate of a set E is the disjoint union of two copies of E, namely E × {a, b}
which has algebra set
k E×{a,b} = k E ⊗ k {a,b} = k E ⊕ k E .
Our purpose in this paper is to concentrate in this case corresponding to noncommutative duplicates of a finite set E.
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We recall first some facts concerning twisted tensor products of algebras and how
they are obtained. Then we provide a classification of the twisted tensor products
of k E and k {a,b} through the set of interlacings which provides an analogous setting
to Ore extensions. The results shows that non-commutative duplicates of a finite
set are provided by one-valued quivers with set of vertices E which has even length
oriented cycles if any and provided with a coloration. The corresponding non semisimple non-commutative 2-nilpotent algebras obtained through the path algebra
construction produces the set of non-commutative duplicates of E.
In the last section we study the Hochschild cohomology of the non-commutative
duplicates of a finite set E. The main result is that the Hochschild cohomology
is finite dimensional if and only if the quiver associated to the situation has no
oriented cycles other than loops, regardless of the coloration. In that case the only
possible non zero vector spaces lies in degrees 0 and 1.
We also compute cyclic homology of this algebras in characteristic zero showing
that the dimension is constant in even degrees. In odd degrees, the dimension of
the cyclic homology counts the number of connected components having a proper
circuit of length dividing the degree augmented by one one.
Concerning Hochschild homology, we obtain that its dimension is finite if and
only if the quiver has no oriented cycles, in which case all the homology vector
spaces vanishes except in degree 0 where the dimension equals the dimension of the
even cyclic homology.
2. Twisted tensor products and interlacings
Let k be a field. We consider k-algebras which are associative and unitary
algebras over k. We recall some basic facts concerning twisted products of algebras.
Let A and B be k-algebras and let A ⊗k B be the tensor product of the underlying k-vector spaces. A twisted tensor product algebra of A and B is a k-algebra
structure on A ⊗k B such that the canonical inclusions a 7→ a ⊗ 1 and b 7→ 1 ⊗ b
are k-algebra maps. Note that we do not assume any extra structure or possible
actions on the original algebras which could provide the algebra structure of the
twisted tensor product. Of course an action or a grading of a group on an algebra
provide examples of twisted products, see 2.6.
Lemma 2.1. Let Λ = A ⊗k B be a twisted tensor product algebra. Then
(a ⊗ 1)(1 ⊗ b) = a ⊗ b.
Proof The product
map m : Λ⊗k Λ → Λ is a Λ-bimodule map and the restriction
to (A ⊗ 1) ⊗ (1 ⊗ B) is an A − B - bimodule map.
Moreover (A ⊗ 1) ⊗ (1 ⊗ B) is a
free A−B bimodule generated by (1⊗1)⊗(1⊗1) and we have m (1⊗1)⊗(1⊗1) =
1 ⊗ 1). Consequently
m (a ⊗ 1) ⊗ (1 ⊗ b) = m a (1 ⊗ 1) ⊗ (1 ⊗ 1) b
= a (1 ⊗ 1) ⊗ (1 ⊗ 1) b
= a m 1 ⊗ 1) b
= a ⊗ b.
As a consequence the structure of a twisted product is determined by the value
of the products of elements of B with elements of A. Let τ : B ⊗ A → A ⊗ B be
NON-COMMUTATIVE DUPLICATES OF FINITE SETS
3
the map given by τ (b ⊗ a) = (1 ⊗ b)(a ⊗ 1), in other words τ provides the product
of b and a inside the twisted tensor product of algebras.
Proposition 2.2. The map τ verifies τ (b ⊗ 1) = 1 ⊗ b, τ (1 ⊗ a) = a ⊗ 1 and the
braiding conditions, namely the diagrams below commute (tensor signs are omitted
or replaced by commas)
BBA


mB ,1A 
y
BA
1B ,τ
−→
BAB
τ,1B
−→
τ
−−−−−−−−−−−−−→
τ,1A
−→
BAA


1B ,mA 
y
ABA
1A ,τ
−→
τ
ABB


 1A ,mB
y
AB
AAB


 mA ,1B
y
BA −−−−−−−−−−−−−→ AB
Proof: The first diagram correspond to the associativity condition of the product for triples (b, b0 , a) namely (b ⊗ 1)(b0 ⊗ 1)(a ⊗ 1) while the second corresponds
to triples (b, a, a0 ).
Definition 2.3. An interlacing is a linear map map τ : BA −→ AB verifying the
preceeding properties.
The following result is now obvious :
Theorem 2.4. Let A and B be k-algebras. The twisted tensor product structures
on A ⊗k B are in one to one correspondence with interlacings.
Example 2.5. The usual tensor product of algebras is obtained through the trivial
interlacing, namely the flip map τ (b ⊗ a) = a ⊗ b.
Example 2.6. Let G be a group acting by automorphisms on a k algebra A and let τ
be the map given by τ (s⊗a) = s(a)⊗s. Then τ is an interlacing which provides the
well known skew group algebra A[G]. As usual this can be generalized, replacing
kG by a Hopf algebra H and the action by an H-module algebra structure on B.
Our purpose is to investigate the set of twisted tensor products between two
algebras in order to provide a classification and properties relatively to the original
algebras. As quoted in the Introduction, in this paper we will consider the case
where B = k {a,b} is the set algebra of two elements {a, b}. Note that the trivial
interlacing provides the usual tensor product A⊗k {a,b} = A⊕A. Note also that the
two elements algebra k {a,b} is isomorphic to the algebra of truncated polynomials
k[X]/(X 2 − X) where a correspond to X and b to 1 − X.
In order to classify interlacings we consider the following set.
Definition 2.7. Let YA be the set of couples (f, δ) such that f is an endomorphism
δ
of A and A −→ fA is an idempotent derivation verifying
f = f 2 + δf + f δ.
Remark 2.8. The notation fA stands for the A-bimodule given by the vector space
A with left action modified by f , namely a · a0 = f (a)a0 . The right action is the
standard one given by the product of A.
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CLAUDE CIBILS
Remark 2.9. The data above can be interpreted as a sort of Ore extensions associated to the quotient of the polynomial algebra in one variable that we consider.
One can prove that Ore extensions (see for instance [7]) are precisely twisted tensor
products of a k-algebra with the polynomial algebra in one variable k[x]. The proof
of this fact can be performed following the lines of the next Proposition.
Proposition 2.10. The set YA is in one-to-one correspondence with the set of
interlacings between the two elements set algebra k[X]/(X 2 − X) and a k−algebra
A.
Proof: Consider the k-module A ⊗ [k[X]/(X 2 − X)] = A[X]/(X 2 − X). An
interlacing τ is determined by the values τ (X, a) corresponding to the product
Xa. We put τ (X, a) = Xa = δ(a) + f (a)X. The braiding conditions provides the
equalities X(Xa) = Xa and X(ab) = (Xa)b. The first one translates into δ 2 = δ
and f = f 2 +δf +f δ while the second provides that f is an endomorphism of A and
δ is a derivation with coefficients in fA. Conversely it is clear that those conditions
insures that the map τ defined by the above formulae is an interlacing. Note that
in this setting the trivial interlacing corresponds to the identity endomorphism and
the zero derivation.
Remark 2.11. One could expect a cohomological interpretation of Ya . It appears
that YA do not seem to have any additional structure, in particular no natural composition law of elements of YA seem to exist. In the next section it will become clear
that interior derivations do not provide in general trivial tensor twisted products.
Definition 2.12. The set of 2-interlacings of an algebra A is the set of interlacings
between A and k[X]/(X 2 − X). We have proved that this set is in one-to-one
correspondence with YA .
3. Set algebras and 2-interlacings
Our first purpose in this section is to describe YkE where k E is the set algebra of
a finite set E. The second purpose is to classify the family of twisted tensor algebras
obtained through the 2-interlacings. Recall that k E is the k-algebra of functions
on E, namely k E = {a : E −→ k} where the product is given by (aa0 )(x) =
a(x)a0 (x). Of course k E is isomorphic to the vector space with basis the set E and
componentwise product
(
X
x∈E
ax x)(
X
x∈E
a0x x) =
X
(ax a0x )x.
x∈E
In other words k E is a product of copies of k indexed by E, and E is a complete
set of primitive orthogonal idempotents of k E .
It is well known that A = k E is cohomologicaly trivial, which means that the
Hochschild cohomology of A with coefficients in any A-bimodule vanish in positive
degrees. In particular any derivation is interior, this fact will provide an exhaustive
way for describing the set of 2-interlacings. For the convenience of the reader as
well as in order to introduce notation we provide a proof of this result.
Let A be a k-algebra and M an A-bimodule. A derivation is a k-linear map
δ : A → M such that δ(aa0 ) = aδ(a0 ) + δ (a)a0 . An interior derivation is associated
to each element m of M by the formula δm (a) = am − ma. One can check that
interior derivations are indeed derivations.
NON-COMMUTATIVE DUPLICATES OF FINITE SETS
5
The following result is well known and easy to prove, it provides the classification
of A-bimodules when A is a finite set algebra over a field.
Proposition 3.1. Let k be a field, E be a set and A = k E be the set algebra. Any
finitely generated A-bimodule is isomorphic to a direct sum of simple modules. The
complete list of simple modules up to isomorphism is {v ku }u,v∈E where v ku is a one
dimensional vector space with identity actions of v on the left, of u on the right,
and zero actions of others set elements.
According to this Proposition it suffices to consider simple bimodule of coefficients in order to prove that any derivation is inner.
Proposition 3.2. Let A = k E be a set algebra, let v ku be a simple bimodule and
let δ : A −→ v ku be a derivation. If u 6= v the derivation is inner. In case u = v
we have δ = 0.
Proof: We prove first that the space of derivations is one-dimensional if u 6= v,
and is zero if u = v. Let e ∈ v ku be a fixed non zero element and for each x ∈ E
let λx ∈ k defined by δ(x) = λx e. We assert that λx = 0 if x 6= u or x 6= v. Indeed,
λx e = δ(x) = δ(x2 ) = xδ(x) + δ(x)x = λx xe + λx ex = 0 + 0 = 0.
If u = v we have λu e = δ(u) = 2λu e which implies λu = 0. If u 6= v then
0 = δ(vu) = vδ(u) + δ(v)u = λu ve + λv eu = (λu + λv )e.
We consider now the interior derivation δe given by δe (x) = xe − ex. Of course we
have δe (x) = 0 if x 6= u or x 6= v. Assuming u 6= v, we obtain δe (v) = e while
δe (u) = −e. Consequently the space of interior derivations is also one dimensional
and every derivation is inner.
Towards the description of the set of 2-interlacings, we recall that each algebra
endomorphism of k E is determined by a set map ϕ : E −→ E. Actually this is
a special case of the anti-equivalence between the category of finite sets and the
category of semi-simple basic and commutative algebras. More precisely let f be
an algebra endomorphism of k E . There exist a unique set map ϕ : E −→ E such
that for each e ∈ E we have
f (e) =
Σ
x.
{x|ϕ(x)=e}
Lemma 3.3. Let f an algebra endomorphism of A = k E given by a set map
ϕ : E −→ E. Let δ : A −→ fA be a derivation. There exist a ∈ A such that for each
e ∈ E we have δ(e) = (f (e) − e)a = Σϕ(x)=e ax x − ae e. Moreover
X
X
δ 2 (e) =
ay aϕ(y) y −
ax (ax + ae )x + a2e e.
ϕ(y)=e
ϕ(x)=e
Remark 3.4. The element a of the Lemma is uniquely determined once normalized
at loop elements of E, namely ae = 0 if f (e) = e. We call if the determining element
of δ.
Proof: Since δ is interior there exist a such that δ(e) = e · a − ae e = f (e)a − ae
= (f (e) − e)a. The fact that a is unique follows at once from the previous considerations. Note also that if u 6= v we have H 0 (A, v ku ) = 0.
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Now
δ 2 (e)
=
X
ax δ(x) − ae δ(e)
ϕ(x)=e

=
X
ax 
ϕ(x)=e
=
X

X
ay y − ax x − ae 
ϕ(y)=x
ay aϕ(y) y −
ϕ(y)=e


X
ax x − ae e
ϕ(x)=e
X
ax (ax + ae )x + a2e e.
ϕ(x)=e
In order to describe the idempotent derivations δ : A −→ fA it is useful to
introduce the quiver of the endomorphism f given by a set map ϕ : E −→ E.
Recall that a quiver Q is a finite oriented graph with set of vertices Q0 , set of
arrows Q1 and two maps s, t : Q1 −→ Q0 providing each arrow with a source and
a target vertex.
Definition 3.5. Let f be an endomorphism of the set algebra A = k E given by a
set map ϕ. The quiver Qf of f has set of vertices E and an arrow from x to ϕ(x)
for each x ∈ E. Two arrows b and a are concatenated if s(b) = t(a)
Remark 3.6. Quivers obtained this way are precisely one-valued quivers, that is
each vertex of the quiver is the source of exactly one arrow, accordingly to the
definition of a set map.
Definition 3.7. An oriented cycle of a quiver is a sequence of concatenated arrow
such that the source of the first arrow coincides with the target of the last one. Its
length is the number of arrows involved. A loop is an oriented cycle of length one,
namely an arrow with the same source and target vertices. An oriented cycle is
proper if it is not the iteration of a strictly smaller length oriented cycle. A loop
vertex is a vertex where a loop has its source and target vertices.
Each connected component of a one-valued quiver has precisely one proper oriented cycle, which can be a loop.
Let R be a connected component of the quiver Qf . Let δ : k E −→ f (k E ) be a
derivation with determining element a ∈ k E . Our first purpose is to describe those
a such that δ 2 = δ.
Lemma 3.8. Let A = k E be a finite set algebra, f an endomorphism of A with set
map ϕ and quiver Qf and let A −→ fA be an idempotent derivation with determining
element a ∈ k E . Let u −→ v be an arrow of Qf with u 6= v where v is a non loop
vertex and such that there is no arrow backwards. Note that u cannot be a loop
vertex since Qf is one-valued. Then au , av ∈ {−1, 0} and au av = 0.
Proof: At the vertex v we have the formula
X
X
0 = (δ 2 − δ)(v) =
ay aϕ(y) y −
ax (ax + av + 1)x + av (av + 1)v.
ϕ2y =v
ϕ(x)=v
The v-coefficient of this sum is av (av + 1), then av ∈ {0, −1}. The u-coefficient is
−au (au + av + 1). If av = −1 then au = 0. If av = 0 then au ∈ {0, −1}.
Next we define pre-colorations of Qf , which will correspond to idempotent derivations.
NON-COMMUTATIVE DUPLICATES OF FINITE SETS
7
Definition 3.9. Let Qf be the quiver of an algebra endomorphism f provided by a
set map ϕ : E −→ E. A pre-coloration of its vertices is an element a of k E verifying
the following conditions:
(1) In case of a connected component of Qf reduced to a round trip quiver
u· ·v the colors au and av verify either au + av + 1 = 0 or au = av = 0.
(2) For a connected component different from the round trip quiver:
(a) In case e is a non loop vertex then ae ∈ {0, −1}.
(b) For each arrow the product of the colors at its source and target is 0.
(c) For a loop vertex the color is irrelevant, we normalize it at 0.
Proposition 3.10. Let A = k E be a finite set algebra, f an endomorphism of A
with set map ϕ and quiver Qf and let A −→ fA be an idempotent derivation with
determining element a ∈ k E . Then a is a pre-coloration of Qf on this connected
component.
Proof: We consider a connected component R of Qf . Assume first there is a
loop vertex e in R and consider the equation (δ 2 − δ)(e) = 0. The coefficient of e is
a2e − ae (2ae + 1) + a2e + ae which is 0 for any value of ae . If the connected component
R is reduced to e we have that a is a pre-coloration. Otherwise let x such that
ϕ(x) = e and x 6= e. The coefficient of x in the equation (δ 2 − δ)(e) = 0 is
ax ae − ax (ax + ae + 1) = ax (ax + 1).
Then ax ∈ {0, −1}. In case R do not contain the round trip quiver the conclusion
follows from the preceding lemma.
If R contains the round trip quiver having vertices u and v the equation (δ 2 −
δ)(u) = 0 provides the v-coefficient
−av (av + au + 1)
and u-coefficient
au av + au (au + 1) = av (av + au + 1).
It follows that au + av + 1 = 0 or au = av = 0.
In case R is not reduced to the round trip quiver there is an arrow arriving to u
or v – we assume u is this vertex without lost of generality – coming from a vertex w
which cannot be a loop vertex since each vertex is the source of exactly one arrow.
For the same reason this arrow cannot be part of a round trip quiver. The Lemma
applies and we infer aw , au ∈ {0, −1} and au aw = 0. If au = 0 the equations of the
round trip quiver above provides av = −1 or av = 0. If au = −1 then av = 0. In
both cases we obtain a pre-coloration.
Remark 3.11. From the proof of the result it is clear that conversely a pre-coloration
of Qf provides an idempotent derivation k E −→ f(k E ).
Finally we need to describe the pre-colorations corresponding to idempotent
derivations verifying f = f 2 + δf + f δ.
P
Definition 3.12. A coloration of Qf is an element a = x∈E ax x ∈ kE such that
(1) For a connected component reduced to the round trip quiver u· ·v the
coefficients au and av verify au + av + 1 = 0.
(2) For other connected components:
(a) In case e is a non loop vertex, ae ∈ {0, −1}
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(b) For each arrow having no loop vertex target, one extremity value is 0
and the other is −1.
(c) At a loop vertex the value of a is irrelevant, we normalise it at 0.
Note that a coloration is a pre-coloration.
Remark 3.13. Not any quiver Qf admits a coloration, clearly the length of non
loop proper oriented cycles has to be even. For a connected component having
even length proper oriented cycles, precisely two colorations exists. Otherwise
a coloration of a connected component with a loop is completely determined by
providing 0 and −1 values on the loop related vertices, namely on source of arrows
having a loop target vertex.
Theorem 3.14. Let E be a finite set, A = k E , and let f be an endomorphism of A
corresponding to a set map ϕ with quiver Qf . Let a ∈ k E be a normalized element
(i.e. it has 0 value at loop vertices) ant let δ : k E −→ f(k E ) be the inner derivation
determined by a. The couple (f, δ) is a 2-interlacing for A if and only if a is a
coloration of Qf .
Proof: We already know that pre-colorations translates the fact that δ 2 = δ.
The condition f = f 2 + δf + f δ provides the following for each e ∈ E :
X
X
(ay + aϕ(y) + 1)y −
(ae + ax + 1)x = 0.
ϕ2 (y)=e
ϕx =e
Let u −→ v be an arrow between different non loop vertices. The u-coefficient
in the above equation for v provides av + au + 1 = 0. Since we have au , av ∈ {0, −1}
we infer that au 6= av . In case v is a loop vertex the u-coefficient in the equation
for v gives
(au + av + 1) − (av + au + 1) = 0
and there no restriction is inferred for av . The v-coefficient for the equation of v
provides
(av + av + 1) − (av + av + 1) = 0.
Finally assume that R is the round trip quiver u v. The u-coefficient for the
v-equation gives
(av + au + 1) = 0
which implies that the option au = av = 0 of a pre-coloration do not provide an
interlacing. In contrast the condition av + au + 1 = 0 is maintained.
Theorem 3.15. Let A = k E be the algebra of a finite set E. The set YA of 2interlacings of A is in bijection with one-valued quivers on the set E provided with
a coloration.
4. Non-commutative duplicates
The aim of this section is to classify the twisted tensor products k E ⊗k {a,b} where
E is a finite set. We will obtain a family of non semi-simple, non-commutative
algebras, with square zero Jacobson radical.
From the preceding section it appears that we can assume that the quiver Qf
of an endomorphism f : k E −→ k E is connected. More precisely let (f, δ) be an
endomorphism of k E and let δ be a derivation
δ : k E −→ f(k E )
NON-COMMUTATIVE DUPLICATES OF FINITE SETS
9
verifying δ 2 = δ and f = f 2 + δf + f δ. Let k E ⊗(f,δ) k[X]/(X 2 = X) be the
corresponding twisted tensor product.
Let E 1 , ..., E n be the partition induced on E by the connected components of
Qf and let fi be the corresponding endomorphism of k Ei . Clearly δ decomposes
into δi : k Ei −→ fi (k Ei ) and we have
Y
k E ⊗(f,δ) k[X]/(X 2 − X) =
k Ei ⊗(fi ,δi ) k[X]/(X 2 − X) .
i=1,··· ,n
First we consider the trivial case of a one element set E, the map f is the identity,
the quiver Qf is a loop and δ = 0. The only twisted tensor product of k E with
k[X]/(X − X 2 ) is the trivial one. There is no non trivial duplicate of a one element
set.
The case where E is an arbitrary finite set but f is the identity reduces to
preceding case, use for instance the connected components of Qf which are all of
them loops, or perform the direct computation since the only derivation is the zero
one.
Recall that for a quiver Q the path algebra kQ has a basis provided by all
the oriented paths – that is finite sequences of concatenated arrows of Q – which
multiplies as they concatenate if this is possible, and have zero product otherwise.
By definition the two sided ideal hQ2 i has a basis given by all the paths of length
greater or equal 2. The quotient k-algebra kQ/hQ2 i has a basis provided by vertices
and arrows. Its Jacobson radical has a basis given by the arrows and has zero
square. It is well known that any finite dimensional, basic and split radical square
zero algebra is obtained this way.
We assert that twisted tensor products k E ⊗ k[X]/(X 2 − X) are members of
the preceding family of algebras. In order to prove this assertion we define a quiver
related to a couple (f, δ) ∈ YkE . It will appear to be essentially the opposite quiver
Qop
f of the set map ϕ defining f , except for loop vertices which becomes two vertices.
Definition 4.1. Let E be a finite set and f : k E −→ k E be an algebra map
provided by a set map ϕ : E −→ E with quiver Qf . Let a be a coloration of Qf
and let δ be the corresponding derivation. The related quiver Q(f,δ) is obtained by
replacing each connected component R of Qf as follows
• If R has no loops replace it by its opposite Rop which has same set of
vertices while arrows are reversed.
• If R has a loop vertex ` (in which case R do not contain other proper
oriented cycles) remove the loop from Rop and replace the vertex ` by two
vertices `0 and `−1 . Let ε = 0 or − 1. Each arrow in Rop from an εpolarized vertex to ` is replaced by an arrow from the ε-polarized vertex to
`ε . Note that this procedure produces two connected components.
Theorem 4.2. Let E be a finite set and let f : k E −→ k E be an algebra endomorphism given by a set map ϕ with a connected quiver Qf . Assume Qf is not the
E
round trip quiver. Let (f, δ) be an interlacing
by a coloration a
of k determined
E
of Qf . The twisted tensor algebra k ⊗(f,δ) k[X]/(X 2 − X) is isomorphic to the
radical square zero algebra kQ(f,δ) /hQ2(f,δ) i.
Remark 4.3. Both algebras have the same dimension since Q(f,δ) has more vertices
but less arrows in the same quantity, namely the loops.
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Proof: Note first that an algebra morphism
ϕ : kQ(f,δ) −→ k E ⊗(f,δ) k[X]/(X 2 − X)
is determined by a coherent choice of images of the vertices and the arrows of Q(f,δ)
since kQ(f,δ) is a tensor algebra on the vertices set algebra of the arrows bimodule.
We define ϕ(e) = e for non loop vertices of Qf . In case of a loop vertex ` of Qf
we put
ϕ(`0 ) = `X and ϕ(`−1 ) = `(1 − X).
F
We have to verify that the set (E \{`}) {`X, `(1 − X)} is a complete set of
orthogonal idempotents. Recall that k E ⊗(f,δ) k[X]/(X 2 − X) is identified with
k E [X]/(X 2 − X) where the product is given by the twist Xb = δ(b) + f (b)X. More
precisely if e ∈ E, we have


X
X
Xe = a(f (e) − e) + f (e)X =
ax x − ae e + 
x X
ϕ(x)=e
ϕ(x)=e
If e is −1-polarised, then
Xe = e +
X
x X.
ϕ(x)=e
If e is 0-polarised


Xe = 
X
x (−1 + X) .
ϕ(x)=e
Finally if e = ` is a loop vertex, then

X
X` = −
x+
ϕ(x)=` ax =−1

X
x X
ϕ(x)=`
Note that

(`X)2 = `X`X = ` −

X
x+
ϕ(x)=` ax =−1
since ϕ(x) = ` and ` belongs to the sum

X

x X  X = `X
ϕ(x)=`
P
ϕ(x)=`
x.
2
Similarly [`(1 − X)] = `(1 − X). We also have that [`X] [`(1 − X)] = 0 =
[`(1 − X)] [`X] since `X`(1 − X) = `X(1 − X) = `X − `X = 0.
Finally note that for a vertex e 6= ` we have e`X = 0 and `Xe = 0 for both
possible colorations of e since
X
1=
e + `X + `(1 − X).
e∈E\{`}
We consider now arrows of Q(f,δ) . For e a non loop vertex and x ∈ E such that
ϕ(x) = e let x ae be the corresponding arrow. We assert that ϕ (x ae ) = xXe is a
coherent choice since ϕ (x ae ) = ϕ (x x ae e) = ϕ(x)ϕ (x ae ) ϕ(e). For ε = 0 or − 1 let
x be a ε-polarized vertex of E such that ϕ(x) = ` and let x a`ε be the corresponding
arrow. We put
ϕ (x a`ε ) = xX`(1 + εX).
NON-COMMUTATIVE DUPLICATES OF FINITE SETS
11
In order to prove that ϕ is surjective note that for a non loop vertex e we have
yXe = 0 if ϕ(y) 6= e. Moreover, if ` is a loop vertex and y is not ε polarized then
yX`(1 + εX) = 0.
Since
X
vXu
X = 1X1 =
u,v vertices of Q(f,δ)
we have that


ϕ
X
a = X.
arrows of Q(f,δ)
E
Of course the entire algebra k is also in the image of ϕ, hence ϕ is surjective. An
easy computation shows that a path of length two of Q(f,δ) has zero image. Since
both algebras have the same dimension the surjective map induces an isomorphism.
Theorem 4.4. Let E = {u, v} and let f : k E −→ k E be an algebra endomorphism
given by a set-map ϕ with quiver Qf = u· ·v. Let (f, δ) be an interlacing of k E
determined by a coloration a = au u + av v of Qf , that is au + av + 1 = 0. The
twisted tensor algebra k E ⊗(f,δ) k[X]/(X 2 − X) is isomorphic to the radical square
zero algebra kQf /hQ2f i.
Remark 4.5. The result of the Theorem is independent of the coloration, in other
words different colorations of the round trip quiver provides the same twisted tensor
product.
Proof: The theory we have developed shows that the twisted tensor product is
the algebra k E [X]/(X −X 2 ) where the interlacing is given by Xu = av v −au u+vX
and Xv = au u − av v + uX. Recall that au + av + 1 = 0. Note also that a direct
computation can be performed to show that the condition is necessary and sufficient
in order to have a well defined associative product.
We assert that the vertices of the algebra are u and v while the arrows are vXu
and uXv. In other words we construct a surjective algebra map ψ : kQf −→
k E [X]/(X − X 2 ). In order to prove this we have to show that X is in the image
of ψ. Indeed X = uXu + vXv + vXu + uXv, the last two terms are in the image
of ψ by construction while uXu = −au u and vXv = −av v. Finally note that
vXuXv = 0 = uXvXu which shows that ψ provides a surjective map between
algebras of dimension 4, consequently they are isomorphic.
It is well known and easy to prove that a complet invariant up to isomorphism
of radical square zero algebras kQ/ < Q2 > is provided by the quiver Q. Of course
this fact will be a crucial piece in order to classify the non-commutative duplicates
of a finite set through the related quiver.
We already note that a connected one-valued quiver not reduced to a loop, which
is not the round trip quiver and with even length oriented cycle has exactly two
colorations. In case there is no loop the related quiver do not depend on the
coloration. In contrast in case there is a loop the related quivers differs depending
on the coloration.
As we quote before, the related quiver of a connected quiver containing a loop
has two connected components. Both have precisely one sink vertex and all other
vertices are one-valued. We call such quivers one-sink-one-valued, observe that a
quiver reduced to a vertex belongs to this family.
This discussion proves the following
12
CLAUDE CIBILS
Theorem 4.6. Let E be a finite set. The complete list up to isomorphism of
non-commutative duplicates of E are provided by the disjoint union of quivers as
follows
• connected one-valued quivers with an even length proper oriented cycle,
• connected one-sink-one-valued quivers
such that the total number of vertices involved equals | E | + L1 where L1 is the
number of connected one-sink-one-valued components (corresponding to the number
of loops of the original quiver). The non-commutative duplicates are the radical
square zero algebras obtained with the opposites of those quivers.
Note that different 2-interlacings can provide the same related quiver. Coloration
switches on non loop connected component provides the same related quiver. In
case the set map has no fixed points (that is the quiver Qf has no loops), the fiber
of the related quiver is precisely described this way. Its cardinality is 2 raised to
the number of such connected components.
5. Hochschild (co)homology and cyclic homology
In order to compute the Hochschild cohomology of non-commutative duplicates
of a finite set, we will use the classification of the preceding section – we have
shown that they are provided by a precise family of quivers and the corresponding
radical square zero algebras. We will also use the main results from [4] where the
dimensions of the Hochschild cohomology for these algebras are computed.
Let Q be a connected quiver provided by a finite set of vertices Q0 , a finite set
of arrows Q1 and two maps s, t : Q1 → Q0 determining the source and the target
of each arrow. Let Qn be the set of paths of length n in Q, namely the sequences
of n concatenated arrows. We denote (kQ)2 the quotient of the path algebra of Q
modulo the two sided ideal generated by Q2 .
The dimension of the Hochschild cohomology of (kQ)2 involves the sets of parallel
paths. More precisely two paths are said to be parallel if they share the same source
and the same terminal vertices. In case X and Y are sets of paths, X//Y is the set
of couples of parallel paths (α, β) where α ∈ X and β ∈ Y .
The following result is proved in [4]. It concerns any connected quiver different
from a crown, where a c-crown is the quiver with c vertices cyclically labelled by
the cyclic group of order c, and c arrows a0 , . . . , ac−1 such that s(ai ) = i and
t(ai ) = i + 1.
Theorem 5.1. Let Q be a connected quiver which is not a crown. Then
dim HH n ((kQ)2 ) = | (Qn //Q1 ) | − | (Qn−1 //Q0 ) | in case n ≥ 2
dim HH 1 ((kQ)2 ) = | (Q1 //Q1 ) | − | Q0 | +1
dim HH 0 ((kQ)2 ) = | (Q1 //Q0 ) | +1
Remark 5.2. In [4] there is a one unity error in the computation of the dimension
of the degree one Hochshchild cohomology.
The preceding Theorem has an important consequence, see Corollary 3.2 of [4]:
NON-COMMUTATIVE DUPLICATES OF FINITE SETS
13
Corollary 5.3. Let Q be a connected quiver which is not a crown. The graded
cohomology HH ∗ ((kQ2 )) is finite dimensional if and only if Q has no oriented
cycles.
Observe that in case of a one-valued quiver the set | (Qn //Q1 ) | for n ≥ 2 is
only originated by oriented cycles. More precisely an oriented cycle of length n − 1
preceded by an arrow provides the set of couples in | (Qn //Q1 ) | for n ≥ 2. Observe
also that in case of a connected one-sink-one-valued quiver | (Qn //Q1 ) |= ∅ for
n ≥ 2.
Theorem 5.4. Let A be a non-commutative duplicate algebra of a finite set, given
by a disjoint union of connected one-sink-one-valued quivers only. Equivalently,
there is no oriented cycles other than loops in the quiver of the set map defining
the non-commutative duplicate.
Then HH n (A) = 0 for n ≥ 2, while dim HH 0 (A) is the number of connected
components of the quiver and dim HH 1 (A) is the Euler characteristic of the underlying graph of Q.
Theorem 5.5. Let A be a non-commutative duplicate algebra of a finite set provided
by a 2-nilpotent algebra associated to a quiver which has an oriented cycle. Then
the graded cohomology HH ∗ (A) is infinite dimensional.
Proof: The result is clear from the previous considerations, and from the fact
that a crown has infinite dimensional cohomology, see Proposition 3.3 of [4].
Finally we resume our results in order to make a direct link with the set YkE of
2-interlacings.
Theorem 5.6. Let (f, δ) be a 2-interlacing of a finite set E, namely f is an endomorphism of k E provided by a set map ϕ : E → E and δ : k E → f(k E ) is
an idempotent derivation verifying f = f 2 + δf + f δ. Let A be the corresponding non-commutative
duplicateof E obtained through the twisting tensor product,
A = k E ⊗(f,δ) k[X]/(X 2 − X) .
Then H ∗ (A) is finite dimensional if and only if the quiver of the set map ϕ do
not contains oriented cycles other than possible loops. In that case the Hochschild
cohomology vanishes in degrees larger than 2.
Proof: The related quiver described in the previous section eliminates loops of
Qf without creating oriented cycles, regardless of the coloration originated by the
derivation.
Concerning cyclic homology we will use the results obtained in [5]. Recall that
the circuit associated to an oriented cycle γ is the orbit of γ through the natural
action of the cyclic group of the same order than the length of γ. The set of circuits
of length j is denoted Θj .
Note that in case of a proper cycle the action of the corresponding cyclic group
is free on its orbit. We denote Ωa the set of proper circuits of length a. The next
result is proven in [5, p. 139]:
Proposition 5.7. Let Q be a quiver and k be a field of characteristic zero.
For n even
dim HCn ((kQ2 )) = | Θn+1 | + | Q0 | .
For n odd
X
dim HCn ((kQ2 )) =
| Ωa | .
2|a|(n+1)
14
CLAUDE CIBILS
Theorem 5.8. Let k be a field of characteristic zero, let E be a finite set, let A be
a non-commutative duplicate algebra of E and let ϕ : E → E be the set algebra map
corresponding to the interlacing. The dimension of the even degree cyclic homology
of A is constant regardless of the derivation:
dim HCeven (A) = | E | + | loops | .
Proof: We know that there are no odd length oriented cycles in the quiver of
f , consequently the dimension is reduced to the number of vertices of the related
quiver. Recall that each loop in the quiver of f produces a new vertex in the related
quiver.
In order to compute the odd cyclic homology, consider a 2-interlacing (f, δ)
and the quiver Qf of f . Let h(a) be the number of connected components of Qf
containing a proper cycle of length a. We know that if a is odd then h(a) = 0.
Theorem 5.9. Let k be a field of characteristic zero, let E be a finite set, let A
be a non-commutative duplicate algebra of E and let f : E → E be the set algebra
map corresponding to the 2-interlacing. Then for n odd
X
dim HCn (A) =
h(a).
a|(n+1)
Finally the Hochschild homology can also be obtained using the computation in
[5, p. 140].
Theorem 5.10. Let k be a field of characteristic different from 2, let E be a finite
set, let A be a non-commutative duplicate algebra of E and let f : E → E be the
set algebra map corresponding to the 2-interlacing.
For n odd
X
dim HHn (A) =
h(a).
a|(n+1)
For n even and n 6= 0
dim HHn (A) =
X
h(a).
a|n
For n = 0
dim HH0 (A) = | E | + | loops | .
References
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344 (1994), no. 2, 885–906.
[2] Cap, Andreas; Schichl, Hermann; Vanžura, Jiřı́ On twisted tensor products of algebras.
Comm. Algebra 23 (1995), no. 12, 4701–4735.
[3] Cartier, Pierre Produits tensoriels tordus. Exposé au Séminaire des groupes quantiques de
l’École Normale Supérieure, Paris, 1991–1992.
[4] Cibils, Claude Hochschild cohomology algebra of radical square zero algebras. Algebras and
modules, II (Geiranger, 1996), 93–101, CMS Conf. Proc., 24, Amer. Math. Soc., Providence,
RI, 1998.
[5] Cibils, Claude Cyclic and Hochschild homology of 2-nilpotent algebras. K-Theory 4 (1990),
no. 2, 131–141.
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[6] Manin, Yu. I. Notes on quantum groups and quantum de Rham complexes. Teoret. Mat. Fiz.
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(1993).
[7] Kassel, Christian Quantum groups. Graduate Texts in Mathematics, 155. Springer-Verlag,
New York, 1995.
Université Montpellier 2, Institut de Mathématiques et de Modélisation de Montpellier, F-34095 Montpellier Cedex 5, France
E-mail address: [email protected]